Which types of networks favor the diffusion of innovations in the sense that an innovation whose intrinsic benefits are greater than those of an established choice will be able to replace the latter when it is initially used only by a small fraction of a large population? For deterministic and regular networks there are characterizations, based on a coordination game model of the diffusion of innovations. Here we study this question for a class of irregular random networks, Small world networks, which are of interest as more realistic models of social networks. We consider a random graph model based on a community structure, in which the choice of a parameter allows us to obtain as special cases several well known models, in particular Watts' Small world. We show that there are different types of Small World graphs some which favor diffusion, others that do not. Our study suggests that the kinds of ties that exist between different communities of an individual play an important role. We interpret Watts' Small World as one with high correlation between social spheres of individuals and favorable to diffusion. In other Small Worlds where the communities of individuals are uncorrelated diffusion succeeds only for very large payoff benefits in favor of the innovation.